Question

What is the average (arithmetic mean) of the first 90 measurements in a certain list of 92 measurements? (1) the average of the 92 measurements is 7.0 centimeters. (2) the average of the last 2 measurements is 7.2 centimeters?

Answer 1

The sum of all the measurements is just the average times the number.

$\bar x = \frac{1}{n} \sum_{i=1}^n x_i$

$\sum_{i=1}^n x_i = n \bar x$

So if the average of all 92 is 7 the sum of those is $92 \times 7$

The average of the last two is 7.2 so their sum is $2 \times 7.2$

That means the sum of the first 90 is $92 \times 7 - 2 \times 7.2$

so the average of the first 90 is

 $\dfrac{92 \times 7 - 2 \times 7.2}{90} = 6.99556$ cm